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Mathematical Modeling II
with Calculus BC

Fractal Geometry

Fractal geometry is the geometry of
chaos theory in the sense that fractal geometry may be used to visually illustrate
the behavior of chaotic dynamical systems. But fractal geometry is much more than this.
It was developed quite apart from chaos theory by a different cast of characters. Its
"father" is indisputably Benoit Mandelbrot who coined the word
"fractal" in 1975, the
same year in which James Yorke gave chaos theory its name. Mandelbrot got part of his inspiration
from the brilliant, but obscure, work of the French mathematician Gaston Julia who, about
sixty years earlier, had developed a theory about iterating functions of complex variables.
Julia could only begin to draw by hand some of the incredible images that would light up
computer screens around the world in the 1980s and 90s, but Mandelbrot would ultimately
bring the expertise of some of IBM's most brilliant computer programmers to bear on Julia's
mathematics.

Mandelbrot had a second source of inspiration from the nineteenth century.
Certain mischievous mathematicians of that era had grown weary of the neverending touting
of the calculus of Newton and Leibniz as the be-all and end-all of mathematics. Do not
misunderstand. It is impossible to overstate the power that the differential and
integral calculus placed in the hands of 18th and 19th century physicists -- it was the
most revolutionary event in the history of mathematics and science. To this day, the
calculus ranks with humankind's greatest intellectual achievements. But the view that
every problem in the real world could be solved by the methods of calculus was wrong.
A parade of 19th century mathematicians -- Weierstrass, Hilbert, Cantor, von Koch,
Sierpinski, and others -- produced fantastic curves that were continuous everywhere,
meaning no "gaps" or "breaks", but differentiable nowhere, meaning essentially that
these curves were all "corners" -- they were not "smooth" anywhere. (For a more detailed
account of two such curves, click here.) The standard curves
of Euclidean geometry -- lines, parabolas, sine waves, and the like -- are all both
differentiable (smooth) and continuous (unbroken), at least almost everywhere. Such curves hold no secrets that
cannot be divined by the power of the calculus. On the other hand, curves which are
everywhere continuous and nowhere differentiable rise up in defiance of any attempt to
be analyzed by Newton's handy set of instructions. In the 19th century these curves
were called "pathological" and "monstrous" and mainstream mathematicians had better things
to do than to waste time dealing with bizarre constructions that had no relevance to
the real world anyway. In the 1970s, Mandelbrot had a different point of view. He saw
in these rough, jagged shapes the potential for
modeling natural objects and phenomena
more accurately than Euclidean geometry ever could. "Mountains are not cones, clouds are
not spheres, bark is not smooth, nor does lightning travel in a straight line," said
Mandelbrot. Far from being pathological, these were the shapes of everday experience.
So Mandelbrot named them "fractals" and wrote the seminal work on fractal geometry called
The Fractal Geometry of Nature in which he introduced
the concept of fractal dimension, a way of characterizing the roughness of fractal shapes.